Hyperbolic paraboloid roof structure and method of constructing the frame thereof



W. S. WHITE, JR ABOLOID ROOF STRUCTURE AND 3,280,518 M THOD OF Oct. 25,1966 HYPERBOLIC PAR 5 Sheets-Sheet 1 Filed Oct. 6, 1959 INVENTOR. MA TEE.5. WH/TE, Je.

i/mWi/m 3,280,518 THOD OF Oct. 25, 1966 w. 5. WHITE, JR

HYPERBOLIC PARABOLOID ROOF STRUCTURE AND ME CONSTRUCTING THE FRAMETHEREOF 5 Sheets-Sheet 2 Filed Oct. 6, 1959 JNVENTOR. M475? 5. WwrE /fe.

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3,280,518 HOD OF W. 8. WHITE, JR

Oct. 25, 1966 HYPERBOLIC PARABOLOI 00F STRUCTURE AND MET CONSTRUCTINGTHE FRAME THEREOF :5 Sheets-Sheet 5 Filed Oct. 6, 1959 IN V EN TOR. M1.752 5. W075, (I.

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United States Patent 3,280,518 HYPERBOLIC PARABOLOID ROOF STRUCTURE ANDMETHOD OF CONSTRUCTING THE FRAME THEREOF Waiter S. White, Jr., PalmDesert, Calif. (R0. Box 310, Colorado Springs, Colo.) Filed Oct. 6,1959, Ser. No. 844,744 9 Claims. (Cl. 52-80) This invention relates tobuilding construction and particularly to a hyperbolic paraboloid roofstructure of the type disclosed and claimed in my copending applicationSer. No. 749,882, filed July 21, 1958 and now abandoned and entitledCurved Roof Structure of which this application is a continuation inpart.

The hyperbolic paraboloid is a geometric configuration defined by astraight line generatrix translated about a transverse normal line andangularly moved about that line as an axis during the course of suchtranslation.

The essential problem encountered in constructing such a structure isthat the horizontal projection of a roof structure is desirablypolygonal, that is, having sides that are straight. Strips of wood ormetal that are uniform in cross-section cannot be used economicallybecause the area described by the generatrix is obviously greater, thefarther away it is from the axis along which the generatrix istranslated.

In my copending application I described and claimed the use of atransversely flexible decking strip made of uniform cross sectionswhereby hand fitting methods may be avoided. The primary object of thisinvention is to improve in general upon the structure as shown in saidcopending application.

In hyperbolic par-aboloid roof structures of the character discussedherein, essentially a two-point suspension or support is provided atdiagonal corners of the structure. The opposite or diagonal corners areelevated. By virtue of the fact that only two points of support arerequired, substantial freedom and versatility of plan design is achievedwithin or beneath the roof structure.

Heretofore the angle of inclination to the horizontal from a point ofsupport to the quadrature point high above the support has been keptrather substantial in order to minimize the cantilever type torqueimposed on the roof. The lower the angle, the greater the torque. Fromthe standpoint of savings in material it is obviously advantageous tomaintain the lowest possible angle to the horizontal. An object of thisinvention is to make possible for the first time a very low aridgraceful angular inclination of the entire roof structure, yetmaintaining the desired two-point suspension.

It has been heretofore understood that there is a substantial stresstending to move the two points of the support outwardly and this is byvirture of the very characteristic of the hyperbolic paraboloidstructure. In other words, the arch that exists in a vertical planepassing through both support points tends, under the weight of the roof,to flatten. This means that the support points of the arch tend to moveapart. An object of this invenion is to provide a new hyperbolicparaboloid roof structure of this character that adds an extra dimensionof strength thereto, whereby no special reinforcements what soever needbe provided in order to restrain the roof from such movement.Accordingly, one of the principal problems heretofore encountered inhyperbolic paraboloid roof structures is now overcome.

In order to achieve the foregoing results use is made of a doubledecking structure, each deck or layer isformed by joining a plurality oftransversely flexible decking elements in side-by-side relationship. Thedecking elements in each layer extend transversely of those of the othervlayer.

3,280,518 Patented Oct. 25, W66

In structuresof this character, it is no small problem to provide anadequate peripheral support for the decking elements. The reason forthis is that the plane of the decking element changes drastically alongthe length of the peripheral support. Thus at the support points of theroof structure, the plane slants downwardly in an outward direction,whereas at the elevated corners, the plane of the roof slants upwardlyin a corresponding direction. In my prior application I utilized tubingat the peripheral supporting structure.

For purposes of providing an appropriately oriented plate for attachmentto the decking elements of the roof, I provided a channel that wastwisted along the length of the tubing, the tubing by its very natureproviding a support appropriately operable irrespective of the angularorientation of the channel about the axis of the tubing. The tubing assuch is not a particularly strong section for present purposes.Furthermore, it is not particularly simple to weld the channel intoposition after the peripheral support member are elevated.

Accordingly, an object of this invention is to provide an improvedmethod of peripheral support whereby the appropriate changingorientation of the support is automatically achieved. For this purpose Iutilize channel elements as the primary peripheral support members, andby the aid of a unique process, cause it to be appropriately oriented byan automatic process as the support members are placed in position.

This invention possesses many other advantages and has other objectswhich may be made more clearly apparent from a consideration of severalembodiments of the invention. For this purpose there are shown a fewforms in the drawings accompanying and forming part of the presentspecification. These forms will now be described in detail, illustratingthe general principles involved; but it is to be understood that thisdetailed description is not to be taken in a limiting sense since thescope of the invention is best defined by the appended claims.

Referring to the drawings:

FIGURE 1 is a pictorial view of a roof structure incorporating thepresent invention;

FIG. is a top plan view thereof;

FIG. 3 is an enlarged fragmentary sectional view taken along a planeindicated by line 3-3 of FIG. 2;

FIG. 4 is an enlarged fragmentary sectional view taken along a planeindicated by line 44 of FIG. 2;

FIG. 5 is an enlarged side elevation of the roof structure;

FIG. 6 is an enlarged sectional view in fragmentary form taken along aplane indicated by line 66 of FIG.

FIG. 7 is a sectional view taken along a plane indicated by line 7-7 ofFIG. 6; 7

FIG. 8 is a diagrammatic view illustrating how the peripheral supportmembers are elevated;

FIG. 9 is a fragmentary side elevation view similar to FIG. 5 butillustrating another embodiment of the present invention;

FIG. 10 is a sectional view similar to FIG. 7, taken along the planeindicated by line 1l10 of FIG. 9, but illustrating an alternateperipheral support member; and

FIG. 11 is a view similar to FIGS. 7 and 10 showing a modified roofstructure in which four layers of decking elements instead of two areprovided. 7

In FIG.--1 thr''i's' illustrated a hyperbolic paraboloid roof structurethat is characterized by the provision of two bearing devices 12 and 14,located at two of the four cornersthereof, upon which the entire roofrests. The corners 16 and 18 of the roof are located at the bearings 12and 14. The opposite corners 20 and 22 (see also FIG. 2) are locatedsubstantially above the bearings 12 and 14. In the present example, thehorizontal projection of the roof structure is a parallelogram otherthan a rectangle. While this is a matter of choice, an equilateralrhombus has a recognized capacity of uniquely fitting with likestructures.

The bearings 12 and 14 each consists of a concrete foundation 11 (FIG.and a generally inverted U- shaped leg 13 made of I-beam or othersuitable structure. The angularity of the central connecting, portions13a conforms to the orientation of the roof at the corners 16 and 18. Agusset plate 17 is interposed between each corner and the leg 13 toprovide an increased area for support and for welding.

The roof comprises four composite peripheral support members 24, 26, 28and 30 (FIG. 2) of virtually identical construction. These peripheralsupport members define a framework between which decking members 32 aresuspended in side by side relationship.

The decking elements are elongate and are uniform in cross section. Atone side of each of the decking elements 32, and as illustrated in FIG.3, a socket 34 is formed and at the opposite side of a depending flange35 is provided. The socket 34 at the right hand side of one deckingelement 32 (FIG. 3) receives the flange 35 of the next adjoining deckingelement 32. The socket and flange arrangement serves as a means wherebythe decking elements can be secured together to form a unitary whole,the-re being suitable welding at the area of the flanges 35 and sockets34. The decking elements are similar to those illustrated in mycopending application.

The decking members by virtue of their longitudinal fluting, aretransversely flexible so that the ends thereof may be bent out and thecentral portions thereof squeezed together, whereby the number ofdecking elements is uniform despite the changing width or dimension ofthe roof. In this connction it must be noted that FIG. 2 represents ahorizontal projection of the roof; in other words, the dimension alongthe roof between the corners 20 and 18 is greater than the dimension ofthe roof along the mid-length indicated by the line 1.

There are two layers of decking elements. The decking elements of theupper layer, as indicated in FIG. 2, extend between the oppositeperipheral support members 24 and 26. The decking elements of the lowerlayer.

extend transversely to the decking elements of the upper layer andbetween the other two peripheral support members 28 and 30. The mannerin which the decking elements 32 are connected to the peripheral supportmembers is illustrated most clearly in FIG. 3, wherein the peripheralsupport member 28 exemplifies the others.

The support member 28 comprises four angles 36, 37, 38 and 39. Theangles 36 and 37 have two of their flanges 36a and 37a in opposedrelationship, while the other two of their other flanges 36b and 37b liein a common plane. Sandwiched between the opposing flanges 36a and 37ais an elongate mounting plate 40 that projects beyond the flanges 36aand 37a and inwardly of the roof. A series of nuts 41 and bolts 42 serveto secure the mounting plate 40 along the length of the support member28. T he ends of the lower decking elements 32 are welded to the underside of the projecting portion of the mounting plate 40. The deckingmembers 32 on the upper side are in asimilar manner secured to the upperside of the mounting plates of the quadrature peripheral support members24 and 26. However, the sidemost element 32 of the upper layer ofdecking members and as illustrated in FIG. 3, is likewise welded to theupper surface of the mounting plate 40.

The unusual additional dimension of strength is added to the structureby virtue of the interponnection of the layers themselves. Thus thetroughs of the'fluted decking elements 32 on the upper layer are securedtothe crests of the decking elements 32 of the lowerlayer. In thisexample, bolts 43 (FIG. 6) are illustrated at those areas where themounting plate 40 is interposed between the layers, and welds as at 45are illustrated at the central areas. The bolts 43 extends as an innerborder along the support members, the welds as at 45 being locatedwithin this border of nuts and bolts.

The roof structure, under load, tends as previously stated, to flatten,the corners 16 and 18 tending to spread apart. As may be appreciated inconnection with FIG. 2, this means that the angularity of theparallelogram structure tends to change, and the layers of deckingmembers tend to rotate with respect to each other. This, in turn, puts ashear on the variou fastening elements 43 and 45. The ability of thefastening elements to resist shear gives the new added strengthdimension to the structure. The interlacing of the decking elements intwo or more layers means that the tendency of the roof to flatten underboth static and dynamic conditions is internally resisted. Meansexternal to the roof resisting movement of the corners 16 and 18 awayfrom each other are not actually required. The corners, however, must beconfined against movements beyond the tolerable limits of deflection ofthe structure. The corners may be anchored to the bearing structures 12and 14 solely for this purpose.

In this case, the roof is permitted to assume whatever position it mayunder its normal loaded conditions.

The bearing structures 12 and 14 are tied together by a reinforcedconcrete beam 66) below the ground level to resist dynamic forceswithout requiring the bearings individually to be overly massive. Ofcourse, a tie rod directly between either pair or roof corners couldaccomplish a similar function, but tie rods so located might not beesthetically pleasing, and might interfere with placement of otherstrucures. The grade beam 60 adds versaility.

The corners 16 and 18 can if desired be anchored so that they are closertogether than they normally would be under their static load. In thiscase, the bearings 12 and 14 and the reinforced beam 69 will then resistboth static and dynamic loading, will be largely in tension.

Since a reinforced concrete beam most effectively is used incompression, this operation may be achieved by spreading the corners 16and 18 apart prior to their anchoring to the bearings 12 and 14.

By tensioning the roof beyond expected dynamic loading it can then beinsured that the beam 60 is always under compression.

Any intermediate distribution of load also is possible. For example, onesatisfactory arrangement to permit half the spreading of the corners asmight result without restraint at the corners. An effective distributionof load between the roof itself and the bearings 12 and 14 therebyresults.

The angle members 38 and 39 are welded in complementary fashion to theangles 36 and 37 to form box-like sections, thereby adding increasedstrength to the peripheral supporting members. The peripheral supportingmember 28, as it extends from anchor corner 16 forms, in essence, one oftwo beams upon which half of the roof is cantilevered.

The peripheral supporting members are joined together at the corners 16,18, 20 and 22. In FIG. 6 a mitered joint is disclosed between thesupport member 28 and the support member 24. The members 24 and 28 arewelded at the mitered joint.

The mounting plates 40 necessarily have changing orientation orinclination to the horizontal along the various peripheral supportmembers 24, 26, 28, 30 in order to parallel those portions of the roofwhich they serve to attach. For this purpose the support members 24, 26,28 and 30 are themselves warped. This may be seen most clearly in aperspective showing of FIG. 1, for example, in 'connection with thesupport of member 24. At the corner 16, a normal to the plane determinedby the flanges 36b and 37b of the support member 24, slopes downwardlyin an'outward direction. The support member 24 is twisted so that at thecorner 22 the opposite angular orientation is perceived.

In order to warp the peripheral supporting members 24, 26, 28 and 30, aunique method is used. First, and as the initial step in theconstruction of the roof proper, the support members are positionedsubtantially at a common level. At each of the corners 16, 18, 2-0 and22, the support members are tied together (FIGS. 6 and 7) by a pair ofstraps 50 and 52. These straps engage the outer surfaces of the flanges36a and 37a of the angles 36 and 37. Half of each strap extends alongthe corresponding support member 24 or 28.

A single bolt 54 for each support member passes through aperture in thestraps 5i and 52 as well as the flanges 35a and 37a, and is held inposition by the aid of a nut 56. The bolts 54 permit slight relativeangular movement of the support members 24 and 28 with respect to eachother about an axis normal to the plane defined by adjacent supportmembers 24 and 28, at least to the extent permitted by slight spacing ofthe support member ends. In this specific example, the axis of movementIn (PEG. 7) may be at either one of the bolts 54. Other devices, such asan actual hinge joint might be provided. The reason why the angularmovement is necessary will appear hereinafter. The straps 5t and 52,while permitting the angular movement described, securely hold theflanges 36a and 37a of both support members in coplanar relationship.Thus the corner joint cannot break apart as by movements of the supportmembers 24 and 28 angularly about an axis n (FIG. 6) lying in the commonplane generally defined by the support members 24 and 28.

After all of the support members 24, 26, 28 and 30 are secured by strapssuch as 50 and 52, the diagonal corners 2G and 22 are moved upwardly.During the course of 1 this movement opposite halves of the roofactually rotate about a horizontal axis extending between the oppositediagonal corners 16 and 18, and which coincides with the axes n thereat.This rotation tends to break the joints at 16 and 18. Since the strapsprevent this movement at all of the corners, the support members Warpautomatically to the desired configuration.

The necessity for permitting angular movement about axes m may beexplained with reference to FIG. 8. As the end of the member 24 forexample, is raised, its horizontal projection tends to shorten towa dthe corner 16, in proportion to the increasing angle of elevation. Themagnitude of shortening is equivalent to a cosine function for theincreasing angle of elevation. The end of the member 3%) likewise tendsto recede, but towards the opposite corner 13. Since the ends of themembers 24 and 30 are secured together, the resultant shortening isalong the symmetry line p. Now it will be apparent that movement of thecorners 26 and 22 together slightly changes the angles at the corners.These angles must be permitted freely to change in order to avoidbowing. Hence the degree of freedom for this purpose is provided.

After the support members are elevated, the members at the corners aresecurely welded together and the straps 5i) and 52 remain to provide anextra measure of strength.

The angles 38 and 39 are positioned on the roof after the angles 36 and37 are elevated as heretofore described.

After the support members are in place, the decking members 32 of thelower layer are next positioned, as by the aid of appropriately locatedframe or scaffolding structure, following generally the contour of theroof. Finally, the lower layer forms a suflicient support for workmen onthe top to place the upper layer in position. As the roof is built layerby layer and row by row, a final rigid construction ultimately resultswhen the roof is completed. During the course of construction, asmaterial is added on the roof, the corners 16 and 18, as indicated inFIG. 5, tend to spread apart and along the upper resting portions of thesupports 12 and 14. Restraints may be provided prior to actual weldingof the corners at a position appropriate for load distribution.

During the course of construction stabilizing rods 64 may be attachedadjacent to the corners 20 and 22 and firmly anchored in order to holdthe roof against tilting about an axis joining the corners 16 and 18,due to a symmetrical loading. The completed roof, however, is quitestable if the load is uniformly distributed. Nevertheless to provide asubstantial safety factor, the stabilizing cables can remain.

In the forms illustratedin FIGS. 9 and 10 a different section for theperipheral support members is provided. In this instance two channels101 and 163 are provided, the webs of which face outwardly in a commonplane. The mounting plate 40 is clamped between the sides of thechannels 101 and 103, substantially in the same manner as in connectionWith the angles 36 and 37 of the form previously described.

The channels 101 and 103 are capable of being twisted or warped readily.In place of other strengthening means, a series of gusset plates 105,107, 109, 111, and 113 are provided, welded successively along the outersurfaces of the channels 191 and 193, and overlying them and each other.The width of the gusset plates conforms to that of the peripheralsupport member. The longest gusset plate 105 is first in contact withthe channels of the support member. This plate 165 extends from thecorner 15 to a place adjacent the corner 22, the end edge 195a of thegusset plate 105 appearing in FIG. 9. Similarly, the next longest gussetplate 107 extends from the corner 16 to a place spaced further from thecorner 22, the edge 107a appearing likewise in FIG. 9. The gusset platesthus provide appropriate reinforcement and increased strength near thebase of what is in essence a cantilever type beam.

In the form illustrated in FIG. 11, four layers of decking elements 120,122, v124 and 126 are provided whereby the inherent strength of the roofmay be further increased. By trapping air in the interstices asubstantial measure of insulation is also provided. Pipes, conduits, andthe like can easily be concealed in the channels of the roof.

The inventor claims:

1. In a hyperbolic paraboloid roof structure: a series of peripheralsupport members joined together to form a closed quadrilateral frame,each of the peripheral support members being elongate and rectilinear,any two contiguous support members defining a plane inclined to theplane defined by the remaining two of the peripheral support members;each peripheral support member comprising two parallel sections havingopposed surfaces, each defined by a straight-line generatrix uniformlyrotated angularly about the longitudinal axis of the correspondmgsupport member as the center of said generatrix is translated along saidaxis; the :generatrix at each position along said longitudinal axisbeing oriented to extend inwardly of the frame and toward correspondingportions of the opposite support member; each peripheral support memberalso having a mounting plate clamped between the said surfaces of saidparallel sections and projecting inwardly of the frame, said mountingplate assuming a configuration corresponding to that of the surfaces ofthe corresponding parallel sections; a first set of decking memberssecured together in side-by-side relationship, and having ends fastenedon the lower sides of the mounting plates at two of the peripheralsupport members; a second set of decking members secured together inside-by-side relationship, and having ends fastened to the upper side ofthe mounting plates at the other two of the peripheral support membersso that the decking members of the second set extend transversely to andare superimposed on the first set of decking members; said decking members being shaped to form a hyperbolic paraboloid structure; andfastening means joining the decking members of the sets of the areas ofcrossing.

2. The combination as set forth in claim 1 in which said decking membersare longitudinally fluted to provide transverse flexibility as well aslongitudinal reinforce ment; in which the end decking members of thefirst set are secured to the lower sides of the mounting plates at saidother two peripheral support members; and in which the end deckingmembers of the second set are secured to the upper sides of the mountingplates at said first two peripheral support members.

3. In a roof structure: elements forming a closed quadrilateral framearrayed with respect to each other to define the boundary of animaginary hyperbolic paraboloid surface, each of the elements of theframe being rectilinear, any two contiguous frame elements defining aplane inclined to the plane defined by the remaining two of the frameelements; the frame being capable of withstanding forces tending toalter the spacing of opposite frame elements; a first set of deckingelements, each having transverse flexibility whereby the width of thedecking element is variable along the length thereof to conform tovarious transverse dimensional requirements, each of said deckingelements having a substantial resistance to longitudinal fiexure; meansjoining the decking elements of said first set in side-by-siderelationship so that the end elements of said first set fall along twoopposite frame elements, the ends of each decking element of said firstset being secured respectively to the other two opposite frame elementsto transmit the tension of said decking elements thereto; a second setof decking elements, each having transverse flexibility whereby thewidth of the decking element is variable along the length thereof toconform to various transverse dimensional requirements, each of saiddecking elements of said second set having a substantial resistance tolongitudinal fiexure; means joining the decking elements of said secondset in side-by-side relationship and in juxtaposed relationship to thedecking elements of the first set so that the end elements of the secondset fall along said other two opposite frame elements, the ends of eachelement of said second set being secured to said first two oppositeframe elements to transmit the tension of said decking elements thereto;and means fastening the decking elements of the sets together at theirareas of crossing to provide a shear type restraining force to assist inmaintaining the decking elements of said sets in the shape of ahyperbolic paraboloid.

4. The combination as set forth in claim 3 together with a pair ofbearings for opposite corners of the frame and fastened to the frame fortransfer of seismic forces of the roof to said bearings.

5. The combination as set forth in claim 4 in which said roof ispre-stressed by altering the nominal working spacing of said corners bysaid fastening means, thereby distributing the seismic load of the roofbetween the decking elements and said bearings.

6. The combination as set forth in claim 5 together 7 with a grade beambetween the bearings, and a floor structure above the grade beam; saidfastening means lengthening the nominal working spacing of said cornerswhereby a nominal compressive load is transmitted to said grade beam.

7. A shell roof structure comprising a plurality of sheets in face toface contact and forming a hyperbolic paraboloid composite structure,each sheet having a plurality of undulations extending along onedimension thereof, one sheet having the undulations thereof at rightangles to the undulations of the contacting sheet, and means rigidlysecuring said contacting sheets at a plurality of points along aplurality of contacting undulations of each sheet for retaining saidsheets together in the shape of a hyperbolic paraboloid.

8. A shell roof structure including a pair of sheets having faces inopposed relationship to each other and forming a hyperbolic paraboloidcomposite structure, each sheet having a plurality of undulationsextending along one dimension thereof, one sheet having the undulationsthereof extending transverse and at a substantial angle to theundulations of the companion sheet, and means rigidly securing saidsheets at a plurality of mutually opposed points along a plurality ofadjacent undulations of each sheet for retaining said sheets togetherwith each sheet in the shape of a hyperbolic paraboloid conforming tothe shape of the other of the sheets.

9. The process of constructing a frame for a hyperbolic paraboloid roofstructure, which comprises: attaching elongated support members togetherto form an open quadrilateral structure while the support members extendsubstantially in a common plane; each of said support members havingparts extending along the length of said support members to formattachment surfaces, said attachment surfaces also extendingsubstantially in a common plane when the support members extendsubstantially in a common plane; holding the members at each of thecorners against angular movement out of the tangent planes at thecorners While permitting angular movement of the members about thecorners in their tangent planes, all while relatively raising two of theop posite corners twisting said support members about their longitudinalaxes thereby automatically producing a longitudinal twist in the supportmembers and causing said attachment surfaces to conform to the edges ofa hyperbolic paraboloid.

References Cited by the Examiner UNITED STATES PATENTS 1,614,334 1/1927Wright 52-48 2,034,383 3/1936 Bonsall 52-49 2,185,274 1/ 1940 Schoeniger52-466 2,245,689 6/1941 Krueger 52-261 2,427,021 9/ 1947 Rapp 52-2222,642,824 6/ 1953 McElhone 52-639 2,654,686 10/ 1953 Hansen 52-4892,754,776 7/1956 Blaski 52-460 2,887,192 5/1959 Schaub 52-488 2,891,4916/ 1959 Richter 52-81 2,912,940 11/1959 Baroni 52-80 FOREIGN PATENTS124,525 9/ 1931 Austria.

781,162 8/1957 Great Britain.

753,204- 8/ 1933 France.

OTHER REFERENCES Architectural Record, p. 72, July 1943. House and HomeMagazine, August 1955, p. 95, 96. House and Home, p. 97, August 1955.

EARL J. WITMER, Primary Examiner. WILLIAM L MUSHAKE, Examiner.

D. W. GRAVES, Assistant Examiner.

8. A SHELL ROOF STRUCTURE INCLUDING A PAIR OF SHEETS HAVING FACES INOPPOSED RELATIONSHIP TO EACH OTHER AND FORMING A HYPERBOLIC PARABOLOIDCOMPOSITE STRUCTURE, EACH SHEET HAVING A PLURALITY OF UNDULATIONSEXTENDING ALONG ONE DIMENSION THEREOF, ONE SHEET HAVING THE UNDULATIONSTHEREOF EXTENDING TRANSVERSE AND AT A SUBSTANTIAL ANGLE TO THEUNDULATIONS OF THE COMPANION SHEET, AND MEANS RIGIDLY SECURING SAIDSHEETS AT A PLURALITY OF MUTUALLY OPPOSED POINTS ALONG A PLURALITY OFADJACENT UNDULATIONS OF EACH SHEET FOR RETAINING SAID SHEETS TOGETHERWITH EACH SHEET IN THE SHAPE OF A HYPERBOLIC PARABOLOID CONFORMING TOTHE SHAPE OF THE OTHER OF THE SHEETS.